Certified finite smooth Fourier approximation and LCU normalization

Set $t=(z-x_0)\mathrm{/}(r+\mathrm{\Delta })$, so $\mathrm{\mid }t\mathrm{\mid }\le 1-\nu$, and $b_k=a_k(r+\mathrm{\Delta }{)}^k$. The omitted Taylor terms have modulus at most ${\sum }_{k\ge K}\mathrm{\mid }{b}_k\mathrm{\mid }(1-\nu {)}^k\le B{e}^{-\nu K}\le e\mathrm{/}16$. Let $u_{2j}=0$ and $u_{2j+1}=(2\mathrm{/}\pi )\left(\genfrac{}{}{0px}{}{2j}{j}\right)\mathrm{/}(4^j(2j+1))$. These are nonnegative and ${\sum }_{\mathrm{\ell }}{u}_{\mathrm{\ell }}=1$, while ${\sum }_{\mathrm{\ell }}{u}_{\mathrm{\ell }}{y}^{\mathrm{\ell }}=(2\mathrm{/}\pi )\arcsin y$ for $\mathrm{\mid }y\mathrm{\mid }<1$. One direct derivation is to solve $(1-y^2)h^{\mathrm{\prime }}(y)=yh(y)$ with $h(0)=1$ by coefficients, obtaining $h(y)={\sum }_j\left(\genfrac{}{}{0px}{}{2j}{j}\right)y^{2j}\mathrm{/}{4}^j=(1-y^2{)}^{-1\mathrm{/}2}$; coefficient integration gives the inverse-sine derivative and its value zero at zero. Letting $y$ increase to one, nonnegativity and finite partial sums give total sum $(2\mathrm{/}\pi )\arcsin 1=1$. Thus for $y=\sin (\pi t\mathrm{/}2)$, $t={\sum }_{\mathrm{\ell }}{u}_{\mathrm{\ell }}{y}^{\mathrm{\ell }}$.

Let $u_{\mathrm{\ell }}^{(k)}$ be the coefficients of the $k$th power of this series, with $u_0^{(0)}=1$. They are nonnegative with sum one. Convolution truncated above $L$ computes their exact coefficients through $L$: all degrees are nonnegative, so discarded high degrees cannot contribute to a lower degree in a later convolution. For $\mathrm{\mid }t\mathrm{\mid }\le 1-\nu$, $\mathrm{\mid }y\mathrm{\mid }\le \cos (\pi \nu \mathrm{/}2)\le e^{-{\nu }^2}$. The last inequality follows by integrating $(\ln \cos v{)}^{\mathrm{\prime }}=-\tan v\le -v$ on $[0,\pi \mathrm{/}2)$ and using ${\pi }^2\mathrm{/}8>1$. Hence the omitted degrees $\mathrm{\ell }>L$ in every power have absolute contribution at most $e^{-{\nu }^{2}L}$. Summing with weights $\mathrm{\mid }{b}_k\mathrm{\mid }$ gives error at most $B{e}^{-{\nu }^{2}L}\le e\mathrm{/}32$.

With $v=\pi t\mathrm{/}2$, the exact identity ${\sin }^{\mathrm{\ell }}v=(i\mathrm{/}2{)}^{\mathrm{\ell }}{\sum }_{j=0}^{\mathrm{\ell }}(-1{)}^j\left(\genfrac{}{}{0px}{}{\mathrm{\ell }}{j}\right)e^{i(2j-\mathrm{\ell })v}$ has coefficient absolute sum one. For the omitted terms $\mathrm{\mid }j-\mathrm{\ell }\mathrm{/}2\mathrm{\mid }>M$, their absolute sum is a fair-binomial tail at most $2e^{-2M^2\mathrm{/}\mathrm{\ell }}$. To prove that bound directly, for a fair-binomial $J$, $\mathbb{E}{e}^{s(J-\mathrm{\ell }\mathrm{/}2)}=\cosh (s\mathrm{/}2{)}^{\mathrm{\ell }}\le e^{\mathrm{\ell }{s}^2\mathrm{/}8}$; the inequality follows termwise from $(2j)!\ge 2^{j}j!$. Markov's inequality with $s=4M\mathrm{/}\mathrm{\ell }$ gives the upper tail $e^{-2M^2\mathrm{/}\mathrm{\ell }}$, and symmetry gives the other tail. The case $\mathrm{\ell }=0$ has no omitted terms. Since $L\le {\nu }^{-2}x+3$ and $M\ge {\nu }^{-1}x+2$, $M^2\ge xL$ (the excess $4{\nu }^{-1}x+4$ dominates $3x$). Thus every retained degree's omitted Fourier tail is at most $2e^{-2x}$. After the coefficient weights the error is at most $2B{e}^{-2x}=e^2\mathrm{/}(512B)\le e\mathrm{/}512$.

Combining the errors gives $e\mathrm{/}16+e\mathrm{/}32+e\mathrm{/}512<e\mathrm{/}8$. Every frequency is $s=2j-\mathrm{\ell }$ with $\mathrm{\mid }j-\mathrm{\ell }\mathrm{/}2\mathrm{\mid }\le M$, so $\mathrm{\mid }s\mathrm{\mid }\le 2M$. Multiplication by $e^{-i\pi s{x}_0\mathrm{/}(2(r+\mathrm{\Delta }))}$ converts the variable $t$ Fourier series to the OP8 evolution $e^{i\pi sz\mathrm{/}(2(r+\mathrm{\Delta }))}$. This phase has modulus one. The exact total coefficient norm is at most ${\sum }_{k<K}\mathrm{\mid }{b}_k\mathrm{\mid }{\sum }_{\mathrm{\ell }\le L}{u}_{\mathrm{\ell }}^{(k)}\le B$: expanding sine powers has norm one, and truncation cannot increase their individual absolute sums. Collisions among signed frequencies can only reduce the norm by the triangle inequality. Rounding the final coefficients, including these phases, with total error at most $e\mathrm{/}16$ adds at most that much uniform error and gives norm at most $B+e\mathrm{/}16$.

Choose a certified normalization $B_{+}\in [B+e\mathrm{/}16,B+e\mathrm{/}8]$ containing the rounded coefficient norm below it; when $B=1$ one may take $B_{+}=1+e\mathrm{/}8$, which is rational only when $e$ is rational. LCU followed by flag dilution has clean compression equal to the rounded Fourier sum divided by $B_{+}$. Its error from $f(H)\mathrm{/}{B}_{+}$ is at most $e\mathrm{/}(4B_{+})$ for a Hermitian $H$ whose spectrum is in the stated interval, by diagonalization. Approximate SELECT evolutions or PREPARE/dilution operations add their declared operator errors through the usual finite product telescoping and weighted block sum. No normalized postselected-state inverse is used.

The largest evolution time is $\pi M\mathrm{/}(r+\mathrm{\Delta })=O({\mathrm{\Delta }}^{-1}\ln (B\mathrm{/}e))$, since $x\ge \ln 64$ and $0<\nu <1$. There are at most $4M+1$ distinct frequencies. The finite convolution arrays need at most $O(K{L}^2+K{L}^2)=O(K{L}^2)$ arithmetic operations (direct truncated convolution, then the binomial coefficients and frequency accumulation); their actual numerical precision and classical bit costs are separate and explicit. Thus this construction proves the scalar and coefficient claims without an uncharged coefficient oracle or a citation-only supplier.